| import OAI.Dynamics.SmoothObstruction.EndpointMass |
|
|
| namespace OAI |
|
|
| section |
|
|
| namespace SmoothObstruction.EndpointFrame |
| open Set Matrix ProductBasis ExteriorStability Filter Topology |
| open scoped ContDiff |
| variable {d : ℕ} |
| local notation "E" => EuclideanSpace ℝ (Fin d) |
| lemma GoodEndpoint.mono {r s : ℕ} {δ ε : ℝ} |
| {b : OrthonormalBasis (Fin d) ℝ E} {a : powersetCard (Fin d) r} |
| {u : Fin r → E} {v : Fin s → E} {f : E → E} {x : E} |
| {D₀ : Matrix (Fin r) (Fin r) ℝ} (h : GoodEndpoint δ b a u v f x D₀) (hδε : δ≤ε) : |
| GoodEndpoint ε b a u v f x D₀ := |
| ⟨h.1,h.2.1,h.2.2.1,fun i j => (h.2.2.2 i j).trans hδε⟩ |
|
|
| /-- A single endpoint minor tolerance for every possible rank in dimension d. -/ |
| theorem four_endpoint_branch_uniform : ∃ δ : ℝ,0<δ ∧ |
| ∀ {p q : ℕ},p≤d → q≤d → |
| ∀ (U V : Submodule ℝ E) (_ : IsCompl U V) |
| (u : OrthonormalBasis (Fin p) ℝ U) (v : OrthonormalBasis (Fin q) ℝ V) |
| (b₀ b₁ c₀ c₁ : OrthonormalBasis (Fin d) ℝ E) |
| (r₀ r₁ : powersetCard (Fin d) p) (s₀ s₁ : powersetCard (Fin d) q) |
| (f₀ f₁ g₀ g₁ : E → E) (x y z w : E) |
| (A₀ A₁ : Matrix (Fin p) (Fin p) ℝ) (D₀ D₁ : Matrix (Fin q) (Fin q) ℝ), |
| GoodEndpoint δ b₀ r₀ (fun i => (u i:E)) (fun i => (v i:E)) f₀ x A₀ → |
| GoodEndpoint δ b₀ r₀ (fun i => (u i:E)) (fun i => (v i:E)) f₀ z A₀ → |
| GoodEndpoint δ b₁ r₁ (fun i => (u i:E)) (fun i => (v i:E)) f₁ y A₁ → |
| GoodEndpoint δ b₁ r₁ (fun i => (u i:E)) (fun i => (v i:E)) f₁ w A₁ → |
| GoodEndpoint δ c₀ s₀ (fun i => (v i:E)) (fun i => (u i:E)) g₀ x D₀ → |
| GoodEndpoint δ c₀ s₀ (fun i => (v i:E)) (fun i => (u i:E)) g₀ w D₀ → |
| GoodEndpoint δ c₁ s₁ (fun i => (v i:E)) (fun i => (u i:E)) g₁ y D₁ → |
| GoodEndpoint δ c₁ s₁ (fun i => (v i:E)) (fun i => (u i:E)) g₁ z D₁ → |
| coordinates b₀ r₀ (f₀ z)=coordinates b₀ r₀ (f₀ x) → |
| coordinates b₁ r₁ (f₁ w)=coordinates b₁ r₁ (f₁ y) → |
| coordinates c₀ s₀ (g₀ w)=coordinates c₀ s₀ (g₀ x) → |
| coordinates c₁ s₁ (g₁ z)=coordinates c₁ s₁ (g₁ y) → |
| ∃ Φ : E×E → E×E,Φ (x,y)=(z,w) ∧ ContDiffAt ℝ ∞ Φ (x,y) ∧ |
| (81/121:ℝ)≤|(fderiv ℝ Φ (x,y)).det| ∧ |
| |(fderiv ℝ Φ (x,y)).det|≤(121/81:ℝ) ∧ |
| ∀ᶠ t in 𝓝 (x,y), |
| coordinates b₀ r₀ (f₀ (Φ t).1)=coordinates b₀ r₀ (f₀ t.1) ∧ |
| coordinates b₁ r₁ (f₁ (Φ t).2)=coordinates b₁ r₁ (f₁ t.2) ∧ |
| coordinates c₀ s₀ (g₀ (Φ t).2)=coordinates c₀ s₀ (g₀ t.1) ∧ |
| coordinates c₁ s₁ (g₁ (Φ t).1)=coordinates c₁ s₁ (g₁ t.2) := by |
| classical |
| have hex (z : Fin (d+1)×Fin (d+1)) := four_endpoint_branch (d:=d) (p:=z.1.val) (q:=z.2.val) |
| choose δ hδ hbranch using hex |
| obtain ⟨β,hβ,hβδ⟩ := SmoothObstruction.finite_positive_bound δ hδ |
| refine ⟨β,hβ,?_⟩ |
| intro p q hp hq U V h u v b₀ b₁ c₀ c₁ r₀ r₁ s₀ s₁ f₀ f₁ g₀ g₁ x y z w A₀ A₁ D₀ D₁ |
| hxF hzF hyF hwF hxP hwP hyP hzP heF heF' heP heP' |
| let j : Fin (d+1)×Fin (d+1) := (⟨p,by omega⟩,⟨q,by omega⟩) |
| exact hbranch j U V h u v b₀ b₁ c₀ c₁ r₀ r₁ s₀ s₁ f₀ f₁ g₀ g₁ x y z w A₀ A₁ D₀ D₁ |
| (hxF.mono (hβδ j)) (hzF.mono (hβδ j)) (hyF.mono (hβδ j)) (hwF.mono (hβδ j)) |
| (hxP.mono (hβδ j)) (hwP.mono (hβδ j)) (hyP.mono (hβδ j)) (hzP.mono (hβδ j)) heF heF' heP heP' |
| end SmoothObstruction.EndpointFrame |
|
|
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric Filter Topology EndpointFrame |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
| local notation "E₀" => EuclideanSpace ℝ (Fin d) |
|
|
| /-- The authentic eight endpoint minors give the local Jacobian germ of the |
| ORIGINAL measurable switch. Rank-zero blocks are included. The agreement is |
| relative to retained original inputs, not an extension or a different law. -/ |
| theorem controlled_original_germ_uniform (hclosed : IsClosed (range I)) |
| (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r C β : ℝ,0<r ∧ 1≤C ∧ 0<β ∧ |
| ∀ {p q : ℕ},p≤d → q≤d → |
| ∀ {A J : Type*} (S : Diffeomorph I I M M ∞) (c : M → ι) |
| (P : M → A) (L : M → J) (R : Set M) |
| (Wf Wb : A×A → Submodule ℝ E₀) (n : ℕ) (ρ δ : ℝ) |
| (Φ : M×M → M×M) (G : Set (M×M)) (x y : M), |
| (∀ z,z∈K (c z)) → (∀ u v,P u=P v → c u=c v) → c x=c y → |
| 0≤ρ → ρ≤r → C*ρ<δ → |
| (∀ u∈G,u.1∈R ∧ u.2∈R ∧ L u.1=L u.2) → |
| (∀ u∈R,∀ v∈R,L u=L v → ∀ z, |
| originalTube I S a c (n:=n) δ u z → originalTube I S.symm a c (n:=n) δ v z → |
| originalEndpoint I S a c (n:=n) (Wf (P u,P (S^[n] u))) u z → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P v,P (S.symm^[n] v))) v z → z=(Φ (u,v)).1) → |
| (∀ u∈R,∀ v∈R,L u=L v → |
| originalTube I S a c (n:=n) δ v (Φ (u,v)).2 ∧ |
| originalTube I S.symm a c (n:=n) δ u (Φ (u,v)).2 ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P v,P (S^[n] v))) v (Φ (u,v)).2 ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P u,P (S.symm^[n] u))) u (Φ (u,v)).2) → |
| (∀ t : Fin (n+1), |
| closedBall ((extChartAt I (a (c (S^[t.val] x)))) (S^[t.val] x)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] x)))) (S.symm^[t.val] x)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S^[t.val] y)))) (S^[t.val] y)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] y)))) (S.symm^[t.val] y)) ρ⊆interior (range I)) → |
| let z := (Φ (x,y)).1 |
| let w := (Φ (x,y)).2 |
| originalTube I S a c (n:=n) ρ x z → originalTube I S.symm a c (n:=n) ρ y z → |
| originalTube I S a c (n:=n) ρ y w → originalTube I S.symm a c (n:=n) ρ x w → |
| let a₀ := a (c x) |
| let X := (extChartAt I a₀) x |
| let Y := (extChartAt I a₀) y |
| let Z := (extChartAt I a₀) z |
| let W := (extChartAt I a₀) w |
| let f₀ := extChartAt I (a (c (S^[n] x))) ∘ S^[n] ∘ (extChartAt I a₀).symm |
| let f₁ := extChartAt I (a (c (S^[n] y))) ∘ S^[n] ∘ (extChartAt I a₀).symm |
| let g₀ := extChartAt I (a (c (S.symm^[n] x))) ∘ S.symm^[n] ∘ (extChartAt I a₀).symm |
| let g₁ := extChartAt I (a (c (S.symm^[n] y))) ∘ S.symm^[n] ∘ (extChartAt I a₀).symm |
| ∀ (U V : Submodule ℝ E₀) (_ : IsCompl U V) |
| (u : OrthonormalBasis (Fin p) ℝ U) (v : OrthonormalBasis (Fin q) ℝ V) |
| (b₀ b₁ c₀ c₁ : OrthonormalBasis (Fin d) ℝ E₀) |
| (r₀ r₁ : powersetCard (Fin d) p) (s₀ s₁ : powersetCard (Fin d) q) |
| (A₀ A₁ : Matrix (Fin p) (Fin p) ℝ) (D₀ D₁ : Matrix (Fin q) (Fin q) ℝ), |
| (∀ j∈r₀,b₀ j∈Wf (P x,P (S^[n] x))) → |
| (∀ v∈Wf (P x,P (S^[n] x)),∀ j∉r₀,b₀.repr v j=0) → |
| (∀ j∈r₁,b₁ j∈Wf (P y,P (S^[n] y))) → |
| (∀ v∈Wf (P y,P (S^[n] y)),∀ j∉r₁,b₁.repr v j=0) → |
| (∀ j∈s₀,c₀ j∈Wb (P x,P (S.symm^[n] x))) → |
| (∀ v∈Wb (P x,P (S.symm^[n] x)),∀ j∉s₀,c₀.repr v j=0) → |
| (∀ j∈s₁,c₁ j∈Wb (P y,P (S.symm^[n] y))) → |
| (∀ v∈Wb (P y,P (S.symm^[n] y)),∀ j∉s₁,c₁.repr v j=0) → |
| GoodEndpoint β b₀ r₀ (fun i => (u i:E₀)) (fun i => (v i:E₀)) f₀ X A₀ → |
| GoodEndpoint β b₀ r₀ (fun i => (u i:E₀)) (fun i => (v i:E₀)) f₀ Z A₀ → |
| GoodEndpoint β b₁ r₁ (fun i => (u i:E₀)) (fun i => (v i:E₀)) f₁ Y A₁ → |
| GoodEndpoint β b₁ r₁ (fun i => (u i:E₀)) (fun i => (v i:E₀)) f₁ W A₁ → |
| GoodEndpoint β c₀ s₀ (fun i => (v i:E₀)) (fun i => (u i:E₀)) g₀ X D₀ → |
| GoodEndpoint β c₀ s₀ (fun i => (v i:E₀)) (fun i => (u i:E₀)) g₀ W D₀ → |
| GoodEndpoint β c₁ s₁ (fun i => (v i:E₀)) (fun i => (u i:E₀)) g₁ Y D₁ → |
| GoodEndpoint β c₁ s₁ (fun i => (v i:E₀)) (fun i => (u i:E₀)) g₁ Z D₁ → |
| coordinates b₀ r₀ (f₀ Z)=coordinates b₀ r₀ (f₀ X) → |
| coordinates b₁ r₁ (f₁ W)=coordinates b₁ r₁ (f₁ Y) → |
| coordinates c₀ s₀ (g₀ W)=coordinates c₀ s₀ (g₀ X) → |
| coordinates c₁ s₁ (g₁ Z)=coordinates c₁ s₁ (g₁ Y) → |
| (∀ᶠ t in 𝓝 (X,Y), |
| P ((extChartAt I a₀).symm t.1)=P x ∧ P ((extChartAt I a₀).symm t.2)=P y ∧ |
| P (S^[n] ((extChartAt I a₀).symm t.1))=P (S^[n] x) ∧ |
| P (S^[n] ((extChartAt I a₀).symm t.2))=P (S^[n] y) ∧ |
| P (S.symm^[n] ((extChartAt I a₀).symm t.1))=P (S.symm^[n] x) ∧ |
| P (S.symm^[n] ((extChartAt I a₀).symm t.2))=P (S.symm^[n] y)) → |
| let e := Prod.map (ChartDensity.coordinates I a₀) (ChartDensity.coordinates I a₀) |
| let s := Prod.map (ChartDensity.inverse I a₀) (ChartDensity.inverse I a₀) |
| ∃ Ψ : E₀×E₀ → E₀×E₀,HasFDerivAt Ψ (fderiv ℝ Ψ (X,Y)) (X,Y) ∧ |
| (81/121:ℝ)≤|(fderiv ℝ Ψ (X,Y)).det| ∧ |
| Ψ =ᶠ[𝓝[ChartDomination.coordinateSet e s G] (X,Y)] (e ∘ Φ ∘ s) := by |
| obtain ⟨r,C,hr,hC,hgerm⟩ := original_pair_germ I hclosed a K hK hKU |
| obtain ⟨β,hβ,hbranch⟩ := four_endpoint_branch_uniform (d:=d) |
| refine ⟨r,C,β,hr,hC,hβ,?_⟩ |
| intro p q hp hq A J S c P L R Wf Wb n ρ δ Φ G x y hc hPc hcxy hρ hρr hρδ hG huniq hsecond hsafe |
| z w ht1 ht2 ht3 ht4 a₀ X Y Z W f₀ f₁ g₀ g₁ U V h u v b₀ b₁ c₀ c₁ r₀ r₁ s₀ s₁ A₀ A₁ D₀ D₁ |
| hb₀ hz₀ hb₁ hz₁ hc₀ hv₀ hc₁ hv₁ hxF hzF hyF hwF hxB hwB hyB hzB he₀ he₁ he₂ he₃ hlabels e s |
| obtain ⟨Ψ,hΨ,hcd,hlo,_,heq⟩ := hbranch hp hq U V h u v b₀ b₁ c₀ c₁ r₀ r₁ s₀ s₁ |
| f₀ f₁ g₀ g₁ X Y Z W A₀ A₁ D₀ D₁ hxF hzF hyF hwF hxB hwB hyB hzB he₀ he₁ he₂ he₃ |
| let r₀' := Prod.map (extChartAt I a₀).symm (extChartAt I a₀).symm |
| have hsrcx : x∈(extChartAt I a₀).source := by simpa only [extChartAt_source] using hKU _ (hc x) |
| have hsrcy : y∈(extChartAt I a₀).source := by |
| dsimp only [a₀]; rw [hcxy]; simpa only [extChartAt_source] using hKU _ (hc y) |
| have hsrcz : z∈(extChartAt I a₀).source := by |
| simpa only [Fin.val_zero,Function.iterate_zero_apply,extChartAt_source] using (ht1 0).1 |
| have hsrcw : w∈(extChartAt I a₀).source := by |
| simpa only [Fin.val_zero,Function.iterate_zero_apply,extChartAt_source] using (ht4 0).1 |
| have hx : r₀' (X,Y)=(x,y) := Prod.ext ((extChartAt I a₀).left_inv hsrcx) ((extChartAt I a₀).left_inv hsrcy) |
| have hz : r₀' (Ψ (X,Y))=(z,w) := by |
| rw [hΨ]; exact Prod.ext ((extChartAt I a₀).left_inv hsrcz) ((extChartAt I a₀).left_inv hsrcw) |
| have hXi : X∈interior (range I) := by |
| simpa only [Fin.val_zero,Function.iterate_zero_apply] using (hsafe 0).1 (mem_closedBall_self hρ) |
| have hYi : Y∈interior (range I) := by |
| dsimp only [Y,a₀]; rw [hcxy] |
| simpa only [Fin.val_zero,Function.iterate_zero_apply] using (hsafe 0).2.2.1 (mem_closedBall_self hρ) |
| have hZi : Z∈interior (range I) := by |
| apply (hsafe 0).1 |
| exact mem_closedBall.mpr (by simpa only [Fin.val_zero,Function.iterate_zero_apply,dist_eq_norm] using (ht1 0).2) |
| have hWi : W∈interior (range I) := by |
| apply (hsafe 0).2.1 |
| exact mem_closedBall.mpr (by simpa only [Fin.val_zero,Function.iterate_zero_apply,dist_eq_norm] using (ht4 0).2) |
| have hct := (ChartDensity.pair_inverse_germ I a₀ (X,Y) |
| ((extChartAt I a₀).map_source hsrcx) ((extChartAt I a₀).map_source hsrcy) hXi hYi).1 |
| have hco : ContinuousAt r₀' (Z,W) := (ChartDensity.pair_inverse_germ I a₀ (Z,W) |
| ((extChartAt I a₀).map_source hsrcz) ((extChartAt I a₀).map_source hsrcw) hZi hWi).1 |
| have hctΨ : ContinuousAt (r₀' ∘ Ψ) (X,Y) := (hΨ ▸ hco).comp hcd.continuousAt |
| have heq' := original_equations_germ I S a c P hPc Wf Wb a₀ x y n b₀ b₁ c₀ c₁ r₀ r₁ s₀ s₁ |
| hb₀ hz₀ hb₁ hz₁ hc₀ hv₀ hc₁ hv₁ Ψ (X,Y) hlabels heq |
| have hag := hgerm S c hc P L R Wf Wb n ρ δ Φ r₀' (r₀' ∘ Ψ) (X,Y) |
| hct hctΨ hρ hρr hρδ (by simpa only [hx] using hsafe) |
| (by simpa only [Function.comp_apply,hx,hz] using ht1) |
| (by simpa only [Function.comp_apply,hx,hz] using ht2) |
| (by simpa only [Function.comp_apply,hx,hz] using ht3) |
| (by simpa only [Function.comp_apply,hx,hz] using ht4) huniq hsecond heq' |
| refine ⟨Ψ,(hcd.differentiableAt (by simp)).hasFDerivAt,hlo,?_⟩ |
| apply ChartDensity.retained_pair_germ I a₀ G Φ Ψ (X,Y) hcd.continuousAt |
| ((extChartAt I a₀).map_source hsrcx) ((extChartAt I a₀).map_source hsrcy) hXi hYi |
| · simpa only [hΨ] using (extChartAt I a₀).map_source hsrcz |
| · simpa only [hΨ] using (extChartAt I a₀).map_source hsrcw |
| · simpa only [hΨ] using hZi |
| · simpa only [hΨ] using hWi |
| · filter_upwards [hag] with t ht |
| intro htG |
| exact ht (hG _ htG).1 (hG _ htG).2.1 (hG _ htG).2.2 |
| end SmoothObstruction.ManifoldCoordinates |
| |
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric Filter Topology EndpointFrame |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
| local notation "E₀" => EuclideanSpace ℝ (Fin d) |
| |
| /-- Fixed old-chart endpoint families give the actual retained-set relative germ. -/ |
| theorem family_original_germ_uniform (hclosed : IsClosed (range I)) |
| (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) : |
| ∃ r C β : ℝ,0<r ∧ 1≤C ∧ 0<β ∧ |
| ∀ {p q : ℕ},p≤d → q≤d → |
| ∀ {A J : Type*} (S : Diffeomorph I I M M ∞) (c : M → ι) |
| (P : M → A) (L : M → J) (R : Set M) |
| (Wf Wb : A×A → Submodule ℝ E₀) (n : ℕ) (ρ δ : ℝ) |
| (Φ : M×M → M×M) (G : Set (M×M)) (x y : M), |
| (∀ z,z∈K (c z)) → (∀ u v,P u=P v → c u=c v) → c x=c y → |
| 0≤ρ → ρ≤r → C*ρ<δ → |
| (∀ u∈G,u.1∈R ∧ u.2∈R ∧ L u.1=L u.2) → |
| (∀ u∈R,∀ v∈R,L u=L v → ∀ z, |
| originalTube I S a c (n:=n) δ u z → originalTube I S.symm a c (n:=n) δ v z → |
| originalEndpoint I S a c (n:=n) (Wf (P u,P (S^[n] u))) u z → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P v,P (S.symm^[n] v))) v z → z=(Φ (u,v)).1) → |
| (∀ u∈R,∀ v∈R,L u=L v → |
| originalTube I S a c (n:=n) δ v (Φ (u,v)).2 ∧ |
| originalTube I S.symm a c (n:=n) δ u (Φ (u,v)).2 ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P v,P (S^[n] v))) v (Φ (u,v)).2 ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P u,P (S.symm^[n] u))) u (Φ (u,v)).2) → |
| (∀ t : Fin (n+1), |
| closedBall ((extChartAt I (a (c (S^[t.val] x)))) (S^[t.val] x)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] x)))) (S.symm^[t.val] x)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S^[t.val] y)))) (S^[t.val] y)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] y)))) (S.symm^[t.val] y)) ρ⊆interior (range I)) → |
| let z := (Φ (x,y)).1 |
| let w := (Φ (x,y)).2 |
| originalTube I S a c (n:=n) ρ x z → originalTube I S.symm a c (n:=n) ρ y z → |
| originalTube I S a c (n:=n) ρ y w → originalTube I S.symm a c (n:=n) ρ x w → |
| let a₀ := a (c x) |
| let X := (extChartAt I a₀) x |
| let Y := (extChartAt I a₀) y |
| ∀ (U V : Submodule ℝ E₀) (_ : IsCompl U V) |
| (u : OrthonormalBasis (Fin p) ℝ U) (v : OrthonormalBasis (Fin q) ℝ V), |
| OriginalEndpointControl I S a c n ρ β (Wf (P x,P (S^[n] x))) (fun i => (u i:E₀)) (fun i => (v i:E₀)) x → |
| OriginalEndpointControl I S a c n ρ β (Wf (P y,P (S^[n] y))) (fun i => (u i:E₀)) (fun i => (v i:E₀)) y → |
| OriginalEndpointControl I S.symm a c n ρ β (Wb (P x,P (S.symm^[n] x))) (fun i => (v i:E₀)) (fun i => (u i:E₀)) x → |
| OriginalEndpointControl I S.symm a c n ρ β (Wb (P y,P (S.symm^[n] y))) (fun i => (v i:E₀)) (fun i => (u i:E₀)) y → |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z → |
| originalEndpoint I S a c (n:=n) (Wf (P y,P (S^[n] y))) y w → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P x,P (S.symm^[n] x))) x w → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P y,P (S.symm^[n] y))) y z → |
| (∀ᶠ t in 𝓝 (X,Y), |
| P ((extChartAt I a₀).symm t.1)=P x ∧ P ((extChartAt I a₀).symm t.2)=P y ∧ |
| P (S^[n] ((extChartAt I a₀).symm t.1))=P (S^[n] x) ∧ |
| P (S^[n] ((extChartAt I a₀).symm t.2))=P (S^[n] y) ∧ |
| P (S.symm^[n] ((extChartAt I a₀).symm t.1))=P (S.symm^[n] x) ∧ |
| P (S.symm^[n] ((extChartAt I a₀).symm t.2))=P (S.symm^[n] y)) → |
| let e := Prod.map (ChartDensity.coordinates I a₀) (ChartDensity.coordinates I a₀) |
| let s := Prod.map (ChartDensity.inverse I a₀) (ChartDensity.inverse I a₀) |
| ∃ Ψ : E₀×E₀ → E₀×E₀,HasFDerivAt Ψ (fderiv ℝ Ψ (X,Y)) (X,Y) ∧ |
| (81/121:ℝ)≤|(fderiv ℝ Ψ (X,Y)).det| ∧ |
| Ψ =ᶠ[𝓝[ChartDomination.coordinateSet e s G] (X,Y)] (e ∘ Φ ∘ s) := by |
| obtain ⟨r,C,β,hr,hC,hβ,hh⟩ := controlled_original_germ_uniform I hclosed a K hK hKU |
| refine ⟨r,C,β,hr,hC,hβ,?_⟩ |
| intro p q hp hq A J S c P L R Wf Wb n ρ δ Φ G x y hc hPc hcxy hρ hρr hρδ hG huniq hsecond hsafe |
| z w ht1 ht2 ht3 ht4 a₀ X Y U V h u v hf₀ hf₁ hb₀ hb₁ he₀ he₁ he₂ he₃ hlabels e s |
| obtain ⟨b₀,r₀,A₀,hbF₀,hzF₀,hmF₀⟩ := hf₀ |
| obtain ⟨b₁,r₁,A₁,hbF₁,hzF₁,hmF₁⟩ := hf₁ |
| obtain ⟨c₀,s₀,D₀,hbB₀,hzB₀,hmB₀⟩ := hb₀ |
| obtain ⟨c₁,s₁,D₁,hbB₁,hzB₁,hmB₁⟩ := hb₁ |
| have hsrc (x : M) : x∈(chartAt H (a (c x))).source := hKU _ (hc x) |
| have hx : x∈(extChartAt I a₀).source := by simpa only [extChartAt_source] using hsrc x |
| have hy : y∈(extChartAt I a₀).source := by dsimp only [a₀]; rw [hcxy]; simpa only [extChartAt_source] using hsrc y |
| have hz : z∈(extChartAt I a₀).source := by |
| simpa only [Fin.val_zero,Function.iterate_zero_apply,extChartAt_source] using (ht1 0).1 |
| have hw : w∈(extChartAt I a₀).source := by |
| simpa only [Fin.val_zero,Function.iterate_zero_apply,extChartAt_source] using (ht4 0).1 |
| have hfx := hmF₀ x (originalTube_self I S a c hρ x hsrc) |
| have hfy := hmF₁ y (originalTube_self I S a c hρ y hsrc) |
| have hbx := hmB₀ x (originalTube_self I S.symm a c hρ x hsrc) |
| have hby := hmB₁ y (originalTube_self I S.symm a c hρ y hsrc) |
| have hfw := hmF₁ w ht3 |
| have hbz := hmB₁ z ht2 |
| rw [←hcxy] at hfy hby hfw hbz |
| exact hh hp hq S c P L R Wf Wb n ρ δ Φ G x y hc hPc hcxy hρ hρr hρδ hG huniq hsecond hsafe |
| ht1 ht2 ht3 ht4 U V h u v b₀ b₁ c₀ c₁ r₀ r₁ s₀ s₁ A₀ A₁ D₀ D₁ |
| hbF₀ hzF₀ hbF₁ hzF₁ hbB₀ hzB₀ hbB₁ hzB₁ |
| hfx (hmF₀ z ht1) hfy hfw hbx (hmB₀ w ht4) hby hbz |
| (originalEndpoint_coordinates I S a c n a₀ x z _ b₀ r₀ hbF₀ hzF₀ hx hz he₀) |
| (originalEndpoint_coordinates I S a c n a₀ y w _ b₁ r₁ hbF₁ hzF₁ hy hw he₁) |
| (originalEndpoint_coordinates I S.symm a c n a₀ x w _ c₀ s₀ hbB₀ hzB₀ hx hw he₂) |
| (originalEndpoint_coordinates I S.symm a c n a₀ y z _ c₁ s₁ hbB₁ hzB₁ hy hz he₃) hlabels |
| end SmoothObstruction.ManifoldCoordinates |
| |
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric EuclideanSplitting ExteriorStability EndpointFrame |
| open scoped ContDiff Matrix.Norms.L2Operator |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| {ι : Type*} [Fintype ι] |
| local notation "E₀" => EuclideanSpace ℝ (Fin d) |
| |
| /-- Both original endpoint families from a SINGLE bilateral gauge. The inverse |
| family uses exactly its reversed gauge, not an unrelated certificate. All |
| constants precede ranks, labels, gauges, and the common starting bases. -/ |
| theorem bilateral_endpoint_control (hclosed : IsClosed (range I)) |
| (S : Diffeomorph I I M M ∞) (a : ι → M) (L : ι → Set M) |
| (hL : ∀ i,IsCompact (L i)) (hLU : ∀ i,L i⊆(chartAt H (a i)).source) |
| (Dg : ℝ) (hDg : 0≤Dg) : |
| ∃ rF K₀F CreF C₁F rB K₀B CreB C₁B : ℝ, |
| 0<rF ∧ 1≤K₀F ∧ 0≤CreF ∧ 0≤C₁F ∧ 0<rB ∧ 1≤K₀B ∧ 0≤CreB ∧ 0≤C₁B ∧ |
| ∀ {p q : ℕ} (c : M → ι) (x : M) (n : ℕ) (b R C δ β γ g : ℝ) |
| (A : BilateralGauge I S a c n (Dg*(b*n)) R x) |
| (T : Finset (Fin d)) (_ : T.card=p) (_ : Tᶜ.card=q), |
| (∀ z,z∈L (c z)) → 0≤b → 1≤b*n → 0≤R → 0≤g → 0≤C → 0≤δ → |
| (∀ j∈T,γ+g≤A.lam j) → (∀ j∉T,A.lam j≤γ-g) → |
| let ρ := Real.exp (-(3/2:ℝ)*SmoothScales.B b*n) |
| K₀F*ρ≤1 → ρ≤rF → K₀B*ρ≤1 → ρ≤rB → |
| (∀ t : Fin (n+1),closedBall ((extChartAt I (a (c (S^[t.val] x)))) (S^[t.val] x)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] x)))) (S.symm^[t.val] x)) ρ⊆interior (range I)) → |
| (∀ B : E₀ →L[ℝ] E₀,‖exterior p B‖≤C*(max 1 ‖B‖)^p) → |
| (∀ B : E₀ →L[ℝ] E₀,‖exterior q B‖≤C*(max 1 ‖B‖)^q) → |
| ∀ (Wf Wb U V : Submodule ℝ E₀),Module.finrank ℝ Wf=p → Module.finrank ℝ Wb=q → |
| (‖(A.forwardSpace I S a c n T).starProjection-Wf.starProjection‖≤1/2) → |
| (‖(A.backwardSpace I S a c n Tᶜ).starProjection-Wb.starProjection‖≤1/2) → |
| (‖(A.forwardSpace I S a c 0 T).starProjection-U.starProjection‖≤δ) → |
| (‖(A.forwardSpace I S a c 0 Tᶜ).starProjection-V.starProjection‖≤δ) → |
| ∀ (u : OrthonormalBasis (Fin p) ℝ U) (v : OrthonormalBasis (Fin q) ℝ V), |
| EndpointBudget.admissible d p Dg R C K₀F CreF C₁F b n g δ β → |
| EndpointBudget.admissible d q Dg R C K₀B CreB C₁B b n g δ β → |
| OriginalEndpointControl I S a c n ρ β Wf (fun i => (u i:E₀)) (fun i => (v i:E₀)) x ∧ |
| OriginalEndpointControl I S.symm a c n ρ β Wb (fun i => (v i:E₀)) (fun i => (u i:E₀)) x := by |
| obtain ⟨rF,K₀F,CreF,C₁F,hrF,hKF,hCrF,hCF,hF⟩ := gauge_endpoint_family I hclosed S a L hL hLU Dg hDg |
| obtain ⟨rB,K₀B,CreB,C₁B,hrB,hKB,hCrB,hCB,hB⟩ := gauge_endpoint_family I hclosed S.symm a L hL hLU Dg hDg |
| refine ⟨rF,K₀F,CreF,C₁F,rB,K₀B,CreB,C₁B,hrF,hKF,hCrF,hCF,hrB,hKB,hCrB,hCB,?_⟩ |
| intro p q c x n b R C δ β γ g A T hTp hTq hc hb hbn hR hg hC hδ hf hs ρ hρF hrF' hρB hrB' |
| hsafe hpolyF hpolyB Wf Wb U V hdimF hdimB hendF hendB hstartF hstartB u v hbudF hbudB |
| constructor |
| · exact hF c x n b R C δ β γ g A T hTp hc hb hbn hR hg hC hδ hf hs hρF hrF' |
| (fun t => (hsafe t).1) hpolyF Wf U V hdimF hendF hstartF hstartB u v hbudF |
| · have hf' (j : Fin d) (hj : j∈Tᶜ) : -γ+g≤(A.reverse I S a c).lam j := by |
| have h := hs j (Finset.mem_compl.mp hj) |
| change -γ+g≤ -A.lam j |
| linarith |
| have hs' (j : Fin d) (hj : j∉Tᶜ) : (A.reverse I S a c).lam j≤ -γ-g := by |
| have h := hf j (by simpa only [Finset.mem_compl,not_not] using hj) |
| change -A.lam j≤ -γ-g |
| linarith |
| apply hB c x n b R C δ β (-γ) g (A.reverse I S a c) Tᶜ hTq hc hb hbn hR hg hC hδ |
| hf' hs' hρB hrB' (fun t => (hsafe t).2) hpolyB Wb V U hdimB |
| · simpa only [BilateralGauge.reverse_forwardSpace] using hendB |
| · simpa only [BilateralGauge.reverse_forwardSpace,BilateralGauge.backwardSpace_zero] using hstartB |
| · simpa only [BilateralGauge.reverse_forwardSpace,BilateralGauge.backwardSpace_zero,compl_compl] using hstartF |
| · exact hbudB |
| end SmoothObstruction.ManifoldCoordinates |
| |
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set EuclideanSplitting |
| open scoped ContDiff Matrix.Norms.L2Operator |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| (S : Diffeomorph I I M M ∞) {ι : Type*} (a : ι → M) (c : M → ι) |
| {n : ℕ} {B R : ℝ} {x : M} (A : BilateralGauge I S a c n B R x) |
| |
| lemma BilateralGauge.forwardSpace_compl (t : ℕ) (T : Finset (Fin d)) : |
| IsCompl (A.forwardSpace I S a c t T) (A.forwardSpace I S a c t Tᶜ) := by |
| have hh := (EuclideanSplitting.splitting A.UF A.VF A.lam |
| (fun t => (A.invF t).1) (fun t => (A.invF t).2) T).ranges_isCompl t |
| rw [EuclideanSplitting.splitting_slow_range,EuclideanSplitting.splitting_fast_range] at hh |
| exact hh |
| |
| /-- The common starting bases are orthonormal within each actual gauge group; |
| no global orthogonality or conditioning of their union is postulated. -/ |
| theorem BilateralGauge.forwardSpace_bases (t : ℕ) (T : Finset (Fin d)) : |
| Nonempty (OrthonormalBasis (Fin T.card) ℝ (A.forwardSpace I S a c t T)) ∧ |
| Nonempty (OrthonormalBasis (Fin (d-T.card)) ℝ (A.forwardSpace I S a c t Tᶜ)) := by |
| have hF : Module.finrank ℝ (A.forwardSpace I S a c t T)=T.card := finrank_groupSpace _ _ _ _ _ |
| have hB : Module.finrank ℝ (A.forwardSpace I S a c t Tᶜ)=d-T.card := by |
| rw [show Module.finrank ℝ (A.forwardSpace I S a c t Tᶜ)=Tᶜ.card from finrank_groupSpace _ _ _ _ _] |
| simp only [Finset.card_compl,Fintype.card_fin] |
| exact ⟨⟨(stdOrthonormalBasis ℝ _).reindex (finCongr hF)⟩, |
| ⟨(stdOrthonormalBasis ℝ _).reindex (finCongr hB)⟩⟩ |
| end SmoothObstruction.ManifoldCoordinates |
| |
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric Filter Topology EndpointFrame |
| open scoped Manifold ContDiff |
| variable {d p q : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] |
| {ι : Type*} |
| lemma OriginalEndpointControl.mono {S : Diffeomorph I I M M ∞} {a : ι → M} {c : M → ι} |
| {n : ℕ} {ρ σ β : ℝ} {W : Submodule ℝ (EuclideanSpace ℝ (Fin d))} |
| {u : Fin p → EuclideanSpace ℝ (Fin d)} {v : Fin q → EuclideanSpace ℝ (Fin d)} {x : M} |
| (h : OriginalEndpointControl I S a c n σ β W u v x) (hρ : ρ≤σ) : |
| OriginalEndpointControl I S a c n ρ β W u v x := by |
| obtain ⟨b,r,D,hb,hr,hg⟩ := h |
| exact ⟨b,r,D,hb,hr,fun y hy => hg y (fun t => ⟨(hy t).1,(hy t).2.trans hρ⟩)⟩ |
| end SmoothObstruction.ManifoldCoordinates |
| |
| namespace SmoothObstruction.ManifoldCoordinates |
| open Set Metric Filter Topology EndpointFrame |
| open scoped Manifold ContDiff |
| variable {d : ℕ} {H : Type*} [TopologicalSpace H] |
| (I : ModelWithCorners ℝ (EuclideanSpace ℝ (Fin d)) H) |
| {M : Type*} [MetricSpace M] [ChartedSpace H M] [IsManifold I ∞ M] |
| [MeasurableSpace M] [BorelSpace M] |
| {ι : Type*} [Fintype ι] |
| local notation "E₀" => EuclideanSpace ℝ (Fin d) |
| |
| omit [MeasurableSpace M] [BorelSpace M] in |
| /-- A genuine pair germ from three bilateral gauges in the ORIGINAL atlas. |
| All radii and minor tolerances precede the finite labels and the fast rank. -/ |
| theorem gauge_original_germ_sharp (hclosed : IsClosed (range I)) |
| (S : Diffeomorph I I M M ∞) (a : ι → M) (K : ι → Set M) |
| (hK : ∀ i,IsCompact (K i)) (hKU : ∀ i,K i⊆(chartAt H (a i)).source) |
| (Dg : ℝ) (hDg : 0≤Dg) : |
| ∃ r Cg β K₀F CreF C₁F K₀B CreB C₁B : ℝ, |
| 0<r ∧ 1≤Cg ∧ 0<β ∧ 1≤K₀F ∧ 0≤CreF ∧ 0≤C₁F ∧ |
| 1≤K₀B ∧ 0≤CreB ∧ 0≤C₁B ∧ |
| ∀ {A J : Type*} (c : M → ι) |
| (P : M → A) (L : M → J) (R : Set M) |
| (Wf Wb : A×A → Submodule ℝ E₀) (n : ℕ) (b R₀ Cext δ ε γ g ρ : ℝ) |
| (Φ : M×M → M×M) (G : Set (M×M)) (x y : M), |
| let σ := Real.exp (-(3/2:ℝ)*SmoothScales.B b*n) |
| (∀ z,z∈K (c z)) → (∀ u v,P u=P v → c u=c v) → c x=c y → |
| 0≤b → 1≤b*n → 0≤R₀ → 0≤Cext → 0≤ε → 0≤g → |
| 0≤ρ → ρ≤σ → σ≤r → Cg*ρ<δ → K₀F*σ≤1 → K₀B*σ≤1 → |
| (∀ p≤d,∀ B : E₀ →L[ℝ] E₀,‖ExteriorStability.exterior p B‖≤Cext*(max 1 ‖B‖)^p) → |
| (∀ p≤d,EndpointBudget.admissible d p Dg R₀ Cext K₀F CreF C₁F b n g ε β) → |
| (∀ p≤d,EndpointBudget.admissible d p Dg R₀ Cext K₀B CreB C₁B b n g ε β) → |
| (∀ u∈G,u.1∈R ∧ u.2∈R ∧ L u.1=L u.2) → |
| (∀ u∈R,∀ v∈R,L u=L v → ∀ z, |
| originalTube I S a c (n:=n) δ u z → originalTube I S.symm a c (n:=n) δ v z → |
| originalEndpoint I S a c (n:=n) (Wf (P u,P (S^[n] u))) u z → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P v,P (S.symm^[n] v))) v z → z=(Φ (u,v)).1) → |
| (∀ u∈R,∀ v∈R,L u=L v → |
| originalTube I S a c (n:=n) δ v (Φ (u,v)).2 ∧ |
| originalTube I S.symm a c (n:=n) δ u (Φ (u,v)).2 ∧ |
| originalEndpoint I S a c (n:=n) (Wf (P v,P (S^[n] v))) v (Φ (u,v)).2 ∧ |
| originalEndpoint I S.symm a c (n:=n) (Wb (P u,P (S.symm^[n] u))) u (Φ (u,v)).2) → |
| (∀ t : Fin (n+1), |
| closedBall ((extChartAt I (a (c (S^[t.val] x)))) (S^[t.val] x)) σ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] x)))) (S.symm^[t.val] x)) σ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S^[t.val] y)))) (S^[t.val] y)) σ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] y)))) (S.symm^[t.val] y)) σ⊆interior (range I)) → |
| let z := (Φ (x,y)).1 |
| let w := (Φ (x,y)).2 |
| originalTube I S a c (n:=n) ρ x z → originalTube I S.symm a c (n:=n) ρ y z → |
| originalTube I S a c (n:=n) ρ y w → originalTube I S.symm a c (n:=n) ρ x w → |
| let a₀ := a (c x) |
| let X := (extChartAt I a₀) x |
| let Y := (extChartAt I a₀) y |
| ∀ (x₀ : M) (Ar : BilateralGauge I S a c n (Dg*(b*n)) R₀ x₀) |
| (Ax : BilateralGauge I S a c n (Dg*(b*n)) R₀ x) |
| (Ay : BilateralGauge I S a c n (Dg*(b*n)) R₀ y) |
| (Tr Tx Ty : Finset (Fin d)),Tx.card=Tr.card → Ty.card=Tr.card → |
| (∀ j∈Tx,γ+g≤Ax.lam j) → (∀ j∉Tx,Ax.lam j≤γ-g) → |
| (∀ j∈Ty,γ+g≤Ay.lam j) → (∀ j∉Ty,Ay.lam j≤γ-g) → |
| Module.finrank ℝ (Wf (P x,P (S^[n] x)))=Tr.card → |
| Module.finrank ℝ (Wf (P y,P (S^[n] y)))=Tr.card → |
| Module.finrank ℝ (Wb (P x,P (S.symm^[n] x)))=d-Tr.card → |
| Module.finrank ℝ (Wb (P y,P (S.symm^[n] y)))=d-Tr.card → |
| ‖(Ax.forwardSpace I S a c n Tx).starProjection-(Wf (P x,P (S^[n] x))).starProjection‖≤1/2 → |
| ‖(Ay.forwardSpace I S a c n Ty).starProjection-(Wf (P y,P (S^[n] y))).starProjection‖≤1/2 → |
| ‖(Ax.backwardSpace I S a c n Txᶜ).starProjection-(Wb (P x,P (S.symm^[n] x))).starProjection‖≤1/2 → |
| ‖(Ay.backwardSpace I S a c n Tyᶜ).starProjection-(Wb (P y,P (S.symm^[n] y))).starProjection‖≤1/2 → |
| ‖(Ax.forwardSpace I S a c 0 Tx).starProjection-(Ar.forwardSpace I S a c 0 Tr).starProjection‖≤ε → |
| ‖(Ay.forwardSpace I S a c 0 Ty).starProjection-(Ar.forwardSpace I S a c 0 Tr).starProjection‖≤ε → |
| ‖(Ax.forwardSpace I S a c 0 Txᶜ).starProjection-(Ar.forwardSpace I S a c 0 Trᶜ).starProjection‖≤ε → |
| ‖(Ay.forwardSpace I S a c 0 Tyᶜ).starProjection-(Ar.forwardSpace I S a c 0 Trᶜ).starProjection‖≤ε → |
| originalEndpoint I S a c (n:=n) (Wf (P x,P (S^[n] x))) x z → |
| originalEndpoint I S a c (n:=n) (Wf (P y,P (S^[n] y))) y w → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P x,P (S.symm^[n] x))) x w → |
| originalEndpoint I S.symm a c (n:=n) (Wb (P y,P (S.symm^[n] y))) y z → |
| (∀ᶠ t in 𝓝 (X,Y), |
| P ((extChartAt I a₀).symm t.1)=P x ∧ P ((extChartAt I a₀).symm t.2)=P y ∧ |
| P (S^[n] ((extChartAt I a₀).symm t.1))=P (S^[n] x) ∧ |
| P (S^[n] ((extChartAt I a₀).symm t.2))=P (S^[n] y) ∧ |
| P (S.symm^[n] ((extChartAt I a₀).symm t.1))=P (S.symm^[n] x) ∧ |
| P (S.symm^[n] ((extChartAt I a₀).symm t.2))=P (S.symm^[n] y)) → |
| let e := Prod.map (ChartDensity.coordinates I a₀) (ChartDensity.coordinates I a₀) |
| let s := Prod.map (ChartDensity.inverse I a₀) (ChartDensity.inverse I a₀) |
| ∃ Ψ : E₀×E₀ → E₀×E₀,HasFDerivAt Ψ (fderiv ℝ Ψ (X,Y)) (X,Y) ∧ |
| (81/121:ℝ)≤|(fderiv ℝ Ψ (X,Y)).det| ∧ |
| Ψ =ᶠ[𝓝[ChartDomination.coordinateSet e s G] (X,Y)] (e ∘ Φ ∘ s) := by |
| classical |
| obtain ⟨rG,Cg,β,hrG,hCg,hβ,hgerm⟩ := family_original_germ_uniform I hclosed a K hK hKU |
| obtain ⟨rF,K₀F,CreF,C₁F,rB,K₀B,CreB,C₁B,hrF,hKF,hCrF,hCF,hrB,hKB,hCrB,hCB,hcontrol⟩ := |
| bilateral_endpoint_control I hclosed S a K hK hKU Dg hDg |
| refine ⟨min rG (min rF rB),Cg,β,K₀F,CreF,C₁F,K₀B,CreB,C₁B, |
| lt_min hrG (lt_min hrF hrB),hCg,hβ,hKF,hCrF,hCF,hKB,hCrB,hCB,?_⟩ |
| intro A J c P L R Wf Wb n b R₀ Cext δ ε γ g ρ Φ G x y σ hc hPc hcxy |
| hb hbn hR₀ hCext hε hg hρ hρσ hσr hρδ hρF hρB hpoly hbudF hbudB |
| hG huniq hsecond hsafe z w ht1 ht2 ht3 ht4 a₀ X Y |
| x₀ Ar Ax Ay Tr Tx Ty hTx hTy hfx hsx hfy hsy hdimFx hdimFy hdimBx hdimBy |
| hendFx hendFy hendBx hendBy hstartFx hstartFy hstartBx hstartBy he₀ he₁ he₂ he₃ hlabels e s |
| let U := Ar.forwardSpace I S a c 0 Tr |
| let V := Ar.forwardSpace I S a c 0 Trᶜ |
| have hcompl : IsCompl U V := Ar.forwardSpace_compl I S a c 0 Tr |
| obtain ⟨⟨u⟩,⟨v⟩⟩ := Ar.forwardSpace_bases I S a c 0 Tr |
| have hp : Tr.card≤d := by simpa using Tr.card_le_univ |
| have hq : d-Tr.card≤d := Nat.sub_le _ _ |
| have hTx' : Txᶜ.card=d-Tr.card := by simp only [Finset.card_compl,Fintype.card_fin,hTx] |
| have hTy' : Tyᶜ.card=d-Tr.card := by simp only [Finset.card_compl,Fintype.card_fin,hTy] |
| have hρrG := hρσ.trans (hσr.trans (min_le_left _ _)) |
| have hρrF := hσr.trans ((min_le_right _ _).trans (min_le_left _ _)) |
| have hρrB := hσr.trans ((min_le_right _ _).trans (min_le_right _ _)) |
| have hcx := hcontrol c x n b R₀ Cext ε β γ g Ax Tx hTx hTx' hc hb hbn hR₀ hg hCext hε |
| hfx hsx hρF hρrF hρB hρrB (fun t => ⟨(hsafe t).1,(hsafe t).2.1⟩) |
| (hpoly _ hp) (hpoly _ hq) (Wf (P x,P (S^[n] x))) (Wb (P x,P (S.symm^[n] x))) U V |
| hdimFx hdimBx hendFx hendBx hstartFx hstartBx u v (hbudF _ hp) (hbudB _ hq) |
| have hcy := hcontrol c y n b R₀ Cext ε β γ g Ay Ty hTy hTy' hc hb hbn hR₀ hg hCext hε |
| hfy hsy hρF hρrF hρB hρrB (fun t => ⟨(hsafe t).2.2.1,(hsafe t).2.2.2⟩) |
| (hpoly _ hp) (hpoly _ hq) (Wf (P y,P (S^[n] y))) (Wb (P y,P (S.symm^[n] y))) U V |
| hdimFy hdimBy hendFy hendBy hstartFy hstartBy u v (hbudF _ hp) (hbudB _ hq) |
| have hsafeρ : ∀ t : Fin (n+1), |
| closedBall ((extChartAt I (a (c (S^[t.val] x)))) (S^[t.val] x)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] x)))) (S.symm^[t.val] x)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S^[t.val] y)))) (S^[t.val] y)) ρ⊆interior (range I) ∧ |
| closedBall ((extChartAt I (a (c (S.symm^[t.val] y)))) (S.symm^[t.val] y)) ρ⊆interior (range I) := by |
| intro t |
| exact ⟨(closedBall_subset_closedBall hρσ).trans (hsafe t).1, |
| (closedBall_subset_closedBall hρσ).trans (hsafe t).2.1, |
| (closedBall_subset_closedBall hρσ).trans (hsafe t).2.2.1, |
| (closedBall_subset_closedBall hρσ).trans (hsafe t).2.2.2⟩ |
| exact hgerm hp hq S c P L R Wf Wb n ρ δ Φ G x y hc hPc hcxy hρ hρrG hρδ |
| hG huniq hsecond hsafeρ ht1 ht2 ht3 ht4 U V hcompl u v |
| (hcx.1.mono I hρσ) (hcy.1.mono I hρσ) (hcx.2.mono I hρσ) (hcy.2.mono I hρσ) he₀ he₁ he₂ he₃ hlabels |
| end SmoothObstruction.ManifoldCoordinates |
| |
| end |
| |
| end OAI |
| |
